Algebraic Structures in Topology

DMS-0104004

Alexander A. Voronov

The goal of the project is to discover and study new algebraic

structures in topology suggested or motivated by mathematical

physics, in particular, quantum field theory and string theory.

More specifically, the project aims at discovering a new algebraic

structure on the homology of an n-sphere space, by which we mean

the space of continuous maps from the n-dimensional sphere to a

given manifold. This part of the project, joint with Dennis

Sullivan, generalizes the work pioneered by Chas and Sullivan

in the case n=1, i.e., that of a usual free loop space. Another

goal is to establish connection between Chas-Sullivan's work and Gromov-Witten invariants, which we believe to be a holomorphic

version of Chas-Sullivan's algebraic structure. Gromov-Witten invariants come from sigma model of quantum field theory, and Chas-Sullivan's work 'String Topology' may be regarded as a

topological version of the physical construction. This part of

the project is suggested to be completed by developing a fusion intersection theory of semi-infinite cycles in infinite

dimensional manifolds. Finally, part of the project is

dedicated to relating the above to Kontsevich's Conjecture,

which generalizes Deligne's Conjecture and unravels a deep

relation between deformation theory of abstract n-algebras

and the topology of configuration spaces of points in an (n+1)-dimensional Euclidean space.

The main idea of Algebraic Topology is to be able to recognize

topological properties of a geometric object by associating

algebraic data or structure to the geometric object. Sometimes

the geometry is too complicated to allow immediate understanding

and work with the object, while the algebraic information is

usually simpler by its nature. This project suggests some new

algebraic structure for a sphere space, the space of maps from

an n-dimensional sphere to a manifold. Such spaces are quite complicated and the classical work of Chen, Segal, Jones,

Getzler, Burghelea, Fedorowicz, Goodwillie, and others, produced

not only the computation of the homology of loop spaces, which

are the particular case of sphere spaces for n=1, but

also revealed amazing connections with algebra (Hochschild

complex). Also, recent progress in string theory emphasized

the importance of invariants associated to holomorphic maps

from the 2-sphere to a manifold (Gromov-Witten invariants).

In this project we undertake an analogous study of continuous

maps from the n-sphere to a manifold, which for n=1 has already

enabled significant progress in topology.